Research On Optimization Problems And Related Problems

In this thesis, we study three related problems: optimization problems, equilibrium problems and complementarity problems. This thesis is divided into five chapters. It is organized as follow:In Chapter 1, we consider the stability and well-posedness of star-shaped vector optimization problems. We introduce three classes of increasing-alongrays maps and investigate the relations between increasing-along-rays property and star-shaped vector optimization problems. We also study the stability and well-posedness issues in the senses of Dentcheva and Helbig, and Huang in star-shaped vector optimization problems associated with increasing-along-rays maps.In Chapter 2, we consider the well-posedness issue of set-valued vector optimization problems. We introduce the concept of extended well-posedness in the sense of Bednarczuk for set-valued vector optimization problems, which is named extended B-well-posedness. This notion of well-posedness can be interpreted as some sort of well-posedness under perturbation in terms of Hausdorff set-convergence. To investigate the extended B-well-posedness, we generalize property (H) due to Miglierina and Molho to set-valued and perturbed case. Under a convexity assumption, we show that the extended B-well-posedness is closely related to property (H).In Chapter 3, we consider the least element problem of the feasible set for an equilibrium problem. We introduce the concepts of a feasible set for an equilibrium problem and of a Z-condition for a bifunction. Under Z-condition and strict pseudomonotonicity assumptions, we establish the equivalences among the equilibrium problem, the least element problem of the feasible set, and a related optimization problem. With an additional growth condition, we further prove that the feasible set of an equilibrium problem is a sublattice.In Chapter 4, we consider the solvability of a system of vector equilibrium problems With (S)_+-conditions. We introduce the concept of (S)_+-conditions for a family of maps, which covers the concepts of existing (S)_+-conditions. We establish some existence theorems for systems of vector equilibrium problems with (S)_+-conditions by using the Kakutani-Fan-Glicksberg fixed point theorem.In Chapter 5, we consider the relations between the feasibility and solvability of a vector complementarity problem. We prove that the vector complementarity problem with a pseudomonotonicity assumption is solvable whenever it is strictly feasible. By using the previous results, we further show that the homogeneous vector complementarity problem is solvable whenever it is feasible. At last, we study the solvability of the feasible vector complementarity problem on product spaces.